A measure of axial symmetry of centrally symmetric convex bodies

Marek Lassak¹ ; Monika Nowicka¹

¹Institute of Mathematics and Physics University of Technology 85-796 Bydgoszcz, Poland

Colloquium Mathematicum, Tome 121 (2010) no. 2, pp. 295-306

Résumé

Denote by $K_m$ the mirror image of a planar convex body $K$ in a straight line $m$. It is easy to show that $K^*_m = {\rm conv}(K\cup K_m)$ is the smallest by inclusion convex body whose axis of symmetry is $m$ and which contains $K$. The ratio ${\rm axs}(K)$ of the area of $K$ to the minimum area of $K^*_m$ over all straight lines $m$ is a measure of axial symmetry of $K$. We prove that ${\rm axs}(K) > {1\over 2}\sqrt 2$ for every centrally symmetric convex body and that this estimate cannot be improved in general. We also give a formula for ${\rm axs}(P)$ for every parallelogram $P$.

Détail

DOI : 10.4064/cm121-2-12

Keywords: denote mirror image planar convex body straight line easy * conv cup smallest inclusion convex body whose axis symmetry which contains ratio axs area minimum area * straight lines measure axial symmetry prove axs sqrt every centrally symmetric convex body estimate cannot improved general formula axs every parallelogram nbsp